www.downloadslide.com 10.7 Some Comments on the Theory of Hypothesis Testing 519 support for the hypothesis we wanted to support—the research hypothesis. If, however, Y does not fall in the rejection region and we can determine no specific value of p in H a that is of direct interest, we simply state that we will not reject H 0 and must seek additional information before reaching a conclusion. Alternatively, we could report the p-value associated with the statistical test and leave the interpretation to the reader. If H 0 is rejected for a “small” value of α (or for a small p-value), this occurrence does not imply that the null hypothesis is “wrong by a large amount.” It does mean that the null hypothesis can be rejected based on a procedure that incorrectly rejects the null hypothesis (when H 0 is true) with a small probability (that is, with a small probability of a type I error). We also must refrain from equating statistical with practical significance. If we consider the experiment described and analyzed in Examples 10.7 and 10.11, the p-value of .0124 is “small,” and the result is statistically significant for any choice of α ≥ .0124. However, the difference between the mean reaction times for the two samples is only .2 second, a result that may or may not be practically significant. To assess the practical significance of such a difference, you may wish to form a confidence interval for µ 1 − µ 2 by using the methods of Section 8.6. Finally, some comments are in order regarding the choice of the null hypotheses that we have used, particularly in the one-sided tests. For example, in Example 10.1, we identified the appropriate alternative hypothesis as H a : p

www.downloadslide.com 520 Chapter 10 Hypothesis Testing Exercises 10.59 Applet Exercise Use the applet Hypothesis Testing (for Proportions) (refer to Exercises 10.9– 10.16) to complete the following. Set up the applet to simulate the results of tests of H 0 : p = .8 versus H a : p >.8, using α = .2 and samples of size n = 30. Click the button “Clear Summary” to erase the results of any previous simulations. a b c d Set the true value of p to .8 and implement at least 200 simulated tests. What proportion of simulations results in rejection of the null hypothesis? Leave all settings at their previous values except change the true value of p to .75. Implement at least 200 simulated tests and observe the proportion of the simulations that led to rejection of the null hypothesis. Repeat, setting the true value of p to .7 and again with the true value of p = .65. What would you expect to happen if the simulation was repeated after setting the true value of p to any value less than .65? Try it. Click the button “Show Summary.” Which of the true p’s used in the simulations resulted in the largest proportion of simulated test that rejected the null and accepted the alternative, H a : p >.8? Does this confirm any statements made in the last paragraph of Section 10.7? Which statement? 10.60 Applet Exercise Refer to Exercise 10.59. Set up the applet to simulate the results of tests of H 0 : p = .4 versus H a : p